Vector Proofs

Prove that lines are parallel or that three points lie on a straight line, and write the closing sentence that earns the mark.

In the exam for Edexcel Further Pure Maths 4PM1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1, IGCSE Additional Maths 0606 · 22 real past-paper questions below

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What a Vector Proof Needs

A vector proof shows a fact about lines using a multiple. Most ask for one of two things, and each ends with one sentence.

To prove	Show
Two lines are parallel	one vector is a number times the other
Three points are on a straight line	parallel and a shared point
A shape is a trapezium	one pair of sides is parallel

Key idea

Finding the multiple is the maths. The last sentence, saying what it means, is a mark on its own.

Proving Three Points Are on a Straight Line

The 5-mark question. Here $O A = 6 a$ , $O B = 6 b$ , C is the midpoint of OA, D is on AB with $A D : D B = 2 : 1$ , and $B E = O B$ . Prove C, D and E are on a straight line.

Worked example

1
$A B = A O + O B$ — First the side that D sits on. Go from A back to O, then out to B.
2
$= - 6 a + 6 b$ — A to O goes against $O A$ , so it is $- 6 a$ . O to B is $6 b$ .
3
$C D = C A + A D$ — Now C to D. Go from C along to A, then from A to D.
4
$C A = 3 a$ — C is the midpoint of OA, so CA is half of $6 a$ .
5
$A D = \frac{2}{3} A B$ — $A D : D B = 2 : 1$ makes 3 parts, and AD is 2 of them.
6
$C D = 3 a + \frac{2}{3} (- 6 a + 6 b)$ — Put in CA and AD.
7
$= 3 a - 4 a + 4 b$ — Multiply out the bracket: $\frac{2}{3} \times (- 6) = - 4$ and $\frac{2}{3} \times 6 = 4$ .
8
$= - a + 4 b$ — Collect the $a$ terms: $3 - 4 = - 1$ .
9
$D E = D B + B E$ — Now D to E. Go from D to B, then from B to E.
10
$= \frac{1}{3} (- 6 a + 6 b) + 6 b$ — DB is the other 1 part of 3, so one third of AB. BE equals OB, which is $6 b$ .
11
$= - 2 a + 2 b + 6 b$ — Multiply out the bracket.
12
$= - 2 a + 8 b$ — Collect the $b$ terms: $2 + 6 = 8$ .
13
$= 2 (- a + 4 b)$ — Take out the common factor 2. The bracket is the same as in CD.
14
$= 2 C D$ — So DE is a multiple of CD: exactly twice it.

Rule

Write: $D E = 2 C D$ , so CD and DE are parallel. They share the point D, so C, D and E lie on a straight line.

Common mistake

Stopping at parallel. Two parallel lines can be side by side. Say they share the point D.

Proving Two Lines Are Parallel

The 3-mark question: prove a shape is a trapezium. Here $O A = 4 a$ , $O B = 4 b$ , C is on OA with $O C = 3 a$ , and D is on OB with $O D = 3 b$ .

Worked example

1
$C D = C O + O D$ — Go from C back to O, then out to D.
2
$= - 3 a + 3 b$ — C to O goes against $O C$ , so it is $- 3 a$ . O to D is $3 b$ .
3
$= 3 (b - a)$ — Take out the common factor 3.
4
$A B = A O + O B$ — Use the same route for AB: from A back to O, then out to B.
5
$= - 4 a + 4 b$ — A to O is $- 4 a$ and O to B is $4 b$ .
6
$= 4 (b - a)$ — Take out the common factor 4.
7
$C D = \frac{3}{4} A B$ — Both have the bracket $(b - a)$ : CD has 3 of it and AB has 4. So CD is three quarters of AB.

Rule

Write: CD is a multiple of AB, so CD is parallel to AB. So ABDC has a pair of parallel sides and is a trapezium.

Common mistake

Saying the sides are equal. CD is three quarters of AB: parallel, but shorter.

Finding an Unknown Number (k)

Some questions leave a number such as $k$ unknown. You find it by matching the $a$ terms and the $b$ terms.

Worked example

1
$O N = b, O R = \frac{1}{3} a + \frac{2}{3} b, N T = 6 a + k b, O T = c O R$ — What we are given. O, R and T are on a straight line, so OT is a multiple $c$ of OR.
2
$O T = O N + N T$ — Find OT a second way: go from O to N, then from N to T.
3
$= b + 6 a + k b$ — ON is $b$ , and NT is given.
4
$= 6 a + (1 + k) b$ — Collect the $b$ terms: $b + k b = (1 + k) b$ .
5
$O T = c (\frac{1}{3} a + \frac{2}{3} b) = \frac{c}{3} a + \frac{2 c}{3} b$ — OT is also $c$ times OR. Multiply out the bracket.
6
$\frac{c}{3} = 6$ — Both lines are OT, so the $a$ terms must match.
7
$c = 18$ — Multiply both sides by 3.
8
$1 + k = \frac{2 c}{3}$ — The $b$ terms must match as well.
9
$1 + k = \frac{2 \times 18}{3} = 12$ — Put in $c = 18$ .
10
$k = 11$ — Take 1 from both sides.
11
$18 \times \frac{1}{3} = 6, 18 \times \frac{2}{3} = 12$ — Check: OT is $6 a + 12 b$ , and $1 + 11 = 12$ . Both match.

Key idea

Two vectors are equal only when the $a$ parts match and the $b$ parts match. The vectors $a$ and $b$ point different ways, so one cannot stand in for the other.

Proving With Number Vectors

Sometimes the vectors are given as numbers. The idea is the same: show that one is a multiple of the other.

Worked example

1
$A B = (35), B = (4, 7), E = (16, 27)$ — What we are given. We must show A, B and E are on a straight line.
2
$A = (4 - 3, 7 - 5) = (1, 2)$ — AB goes from A to B, so A is B minus AB. Take 3 from the x and 5 from the y.
3
$A E = (16 - 1 27 - 2) = (1525)$ — End minus start gives the vector from A to E.
4
$= 5 (35)$ — Take out the common factor 5: $15 = 5 \times 3$ and $25 = 5 \times 5$ .
5
$A E = 5 A B$ — The bracket is AB, so AE is 5 times AB.

Rule

Write: $A E = 5 A B$ , so AE is parallel to AB. They share the point A, so A, B and E lie on a straight line.

Common mistake

Taking the start from the end the wrong way round. A vector is always end minus start: $E - A$ , not $A - E$ .

Which Method? Read the Question

Look for the word prove, then what it asks you to prove.

You see	You do
Prove parallel	show one vector is k times the other
Prove a straight line	parallel and a shared point
Prove a trapezium	find one parallel pair of sides
A number $k$ to find	match the $a$ terms, then the $b$ terms
Vectors given as numbers	end minus start, then look for a multiple
Find a ratio where lines cross	Vector Geometry
Is it a multiple?	Multiplying a Vector by a Scalar

Key idea

Take out a common factor from each vector. If the brackets match, you have your multiple.

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